TAP CHi KHOA HOC D H S P TPHCM Sd 9(75) nam 2015
SUY LUAN NGOAI SUY VA QUY NAP TRONG KHAM P H A QUY LUAT DAY SO - N H L T N G PHAN TICH LI THUYET VA THlTC NGHIEM
TRU-ONG TH] KHANH PHUONG*
T O M TAT
Khdm phd quy ludt ddy sd hd tra hgc sinh phdt trien ndng luc suy ludn todn hgc vd viec hieu cdc khdi niem hdm sd vd bien .sd (NCTM, 2000). Bdi bdo ndy phdn tich ca sd li thuyet cho thdy hai logi suy ludn dugc su dung de khdm phd quy ludt ddy so Id ngoai suy vd quy ngp. Kit qud thuc nghiim phdn dnh khd khdn cua hgc sinh trong viec dua ra mgt gid thuyit ngoai suy dii mgnh de hd trg cho quy ngp nhdm di den mgt quy tdc tdng qudt.
Cdc phuang dn ngogi suy dua tren viec khdm phd biiu dien true quan mo td quy ludt ddy sd cd the khac phuc vdn di ndy.
Tie khda: quy luat day so, tong quat hda, suy luan ngoai suy, suy luan quy nap.
ABSTRACT
Abductive reasoning and inductive reasoning in discovering sequence patterns - some theoretical and empirical analysis
Discovering sequence patterns supports students to develope their reasoning and their conceptual understanding of functions and variables (NCTM, 2000). This paper shows that abductive reasoning and inductive reasoning are used to explore sequence patterns. The analysis of data shows that students have difficulties in suggesting a strong abduction that can combine with induction to gel an algebraic rule of sequence pattern.
Abduction based on visual representation which describes the sequence pattern can overcome this problem.
Keywords: sequence pattems, generalization, abductive reasoning, inductive reasoning.
1. Gidi thieu
Polya cho rang toan hpc tdn tai hai kieu suy luan: suy luan diln dich va suy luan cd Ii. Polya nhan manh mdi lien he chat che giiia suy luan dien dich va suy luan cd li nhu sau: "Toan hpc dugc xem la mot mdn khoa hgc chiing minh, tuy nhien dd chi la mpt khia canh cCia nd... Chung ta can phai du doan ve mdt djnh li loan hpc trudc khi chimg minh no, phai du doan ve dudng ldi va tu tudng chii dao ciia chiing minh trudc khi chiing minh, can phai doi chilu cac k i t qua quan sat dugc va suy ra nhimg dieu luong lu, phai md mam va thu di thu lai nhilu lin... Neu viec day toan phan anh d miic dp nao do viec hinh thanh toan hpc nhu t h i nao thi trong viec giang day dd phai danh cho cho du doan, cho suy luan cd li"^. Trong nghien ciiu nay, chung tdi de cap d i n hai
' NCS, Trudng Oai hoc Su" pham TPHCM; Email: [email protected] 106
TAP CHi KHOA HOC DHSP TPHCM Truffng Thi Khanh Phutrng
loai suy luan cd li la suy luan quy. nap va suy l u ^ ngoai suy vi chiing lien quan true tiep din boat ddng kham pha quy luat day sd nhu se trinh bay d phin sau.
2. Suy luan quy nap va suy lu$n ngoai suy 2.1. Suy lugn quy ngp
Suy luan quy nap la lien trinh bat diu tir cac quan sat ve tinh chit/dSc trung cua mdt sd trudng hgp din kit luan vl su tin tai ciia tinh chit do cho mot nhom Idn hon cac trudng hgp ([6]).
Khdng gidng nhu suy luan dien dich, kit luan ciia suy luan quy nap khdng chic ehan diing. Miirc dp cd li ciia ket luan se dugc tang len khi cd nhilu trudng hgp hon dugc kiem chiing la dung, nhung ket luan cd the ngay lap tiic bi bac bd khi cd mgt phan VI du dugc chi ra.
2.2. Suy lugn ngogi suy
Mac dil khai niem vl ngoai suy dugc dl cap den lan diu lien bdi Aristole, nhung nha loan hpc, trilt hgc va logic hpc ngudi MT Charles Sanders Peirce (1839-1914) la ngudi da phat trien khai niem nay va dua nd vao trong he thong cac loai suy luan. Nhan thiic truyen thdng lien quan den ban chat ciia suy luan loan hpc van giii quan diem rang suy dien va quy nap hinh thanh nen rapt cap ddi ma tat ca cac loai suy luan khong phai la suy dien thi se roi vao trudng hgp cdn lai la quy nap [7]. Tuy nhien, Peirce de xuat mpt loai suy luan mdi: suy luan ngoai suy nham tim kiem gia thuyet de li giai cho cac su kien quan sat dugc.
Mo hinh suy luan ngoai suy ciia Peirce:
Mgt sy that C dugc quan sat, NIU A diing, C hiln hien ciing se diing;
Vi thi, la hgp li khi gia thuylt rang A la diing. ([6], 5.189)
J. Josephson & S. Josephson [6] phat trien md hinh ngoai suy cua Peirce them mdt giai doan: danh gia gia thuylt nao la tdt nhat. Md hinh mdi nhim dua ra mdt gia thuylt ngoai suy du tdt dugc viet lai nhu sau:
D \k mpt tap cac dO: lieu (su kien, quan sat, cai da cho) (1) H giai thich D (neu H diing, se giai thich D) (2) Khdng cd gia thuyet khac cd the giai thich D lot hon H (3)
Nhu vay, H cd le 1^ diing. (4) Tinh cd li ciia cac gia thuyet ngoai suy cd the dugc tang len hay bj giam di, tham chi bi
bac bd khi cd them cac su ki?n/thdng lin mdi dugc cung cip. Khi cd nhilu gia thuylt cd li cung giai thich cho mgt quan sat, nhiem vu ciia ngoai suy la chgn ra gia thuyet co ii nhat.
TAP CHI KHOA HOC DHSP TPHCM So 9(75) ndm 2015
2.3. Suy ludn trong qud trinh khdm phd quy lu^t ddy sd a) Cdc nhiem vu khdm phd quy ludt ddy sd
Cac nhiem vu kham pha quy luat day so cd thi dugc thi hien trong bdi canh so hpc (Nhiem vu 1) hay sir dung cac bilu diln tryc quan (BDTQ) dl md ta (Nhiem vu 2) nhu minh hpa dudi day:
Nhiem vu 1. Nam sd hang dau tien ciia mgt day sd la: 1,4,7, 10, 13...
a) Vilt tilp sd hang thii 6, thir 10 va thii 50 cua day sd.
b) Em cd thi vilt mpt quy tic de tim kiem sd hang thir n cua day sd nay nlu gia tri n dugc cho san? Giai thich cac em tim ra cau tra ldi.
Nhiem vu 2. Cac tim bia hinh vuong dugc sap xep thanh cac Mnh chir T theo mpi so do CO quy luat nhu sau:
D
Col
u Tr'T'-n
Cd2 Cd3 Cd4 Hinh 1. Bieu dien true quan md td quy ludt hinh chir T
a) Viet mpt cdng thiic de tim so lam bia can sir dung cho hinh chir T cd «.
b) Sii dung cdng thiic d cau a), tim sd tam bia can dugc sir dung cho hinh chS Ted 100, cd 178.
Trong nhiem vu 1, viec tim kiem sd hang thu 6 ddi hdi hpc sinh (HS) phai xem xet 5 sd hang dugc cho trudc dd va suy ra mpt quy luat giira 5 sd hang nay, chang h ^ : mdi s6 hang diing sau bing sd hang liln kl trudc cdng 3 don vi. Do do, sd hang thii 6 la 13 + 3 = 16. Viec tim kilm sd hang thii 10 va 50 dugc ggi la cac nhiem vutdng quat hda (TQH) gan va TQH xa. TQH gan yeu cau HS tim kiem mpt sd hang khdng han phai lien ke ngay sau cac sd hang da cho, nhung vi tri cua nd trong day quy luat du gan de HS cd the thuc hien timg budc dem tuan ty va co dugc cau tra ldi. TQH xa yeu cau HS tim kiem mgt sd hang d vi tri xa hon nhieu so vdi cac sd hang da dugc cho san khien cho viec dem timg budc tuan ty trd nen khdng hieu qua. Tuy nhien vdi nhiem vu 2, HS khong can phai tien hanh bit ki TQH xa nao dh dat dugc cau tra ldi cho cau hoi b) va c) ma cau tra ldi cho mot vj tri bit ky co thi suy ra ngay tir quy luat dugc thilt lap d cau hdi a). Cac nhiem vu kham pha quy luat day sd ma chiing tdi sii dung trong nghien ciiu nay dugc md ta bing BDTQ.
TAP CHi KHOA HOC DHSP TPHCM Truang Thi Khanh Phwomg
b) Suy ludn cd li trong khdm phd quy ludt day sd
Khi de cap den nhung suy luan xay ra dua tren viec quan sat mdt sd trudng hgp cu the den mgt kit qua long quat, ngudi ta thudng nghT din suy luan quy nap. Khai niem ngoai suy cung khdng he dugc nhac den trong nhiing phan tich cua cac tac gia Reid [9], Canadas & Castro [4] vl suy luan cua HS khi thyc hien cac nhiem vu kham pha quy luat day sd. Tuy nhien, viec ddng nhat nhiem vu kham pha quy luat day sd vdi hanh dgng kiem chiing va tdng quat hda mdt quy luat tir cac trudng hgp cho sin ciia day sd dudng nhu da phdt Id di yeu to sang tao trong qua trinh nay, ylu td ma Peirce da chi ra nhu mpt dac tnmg cua ngoai suy. Trong khi do, Canadas va Castro khing dinh ring trong sd bay budc ciia lien trinh suy luan quy nap bao gdm: (1) Quan sdt cdc trudng hgp dgc biet; (2) Sdp xep cdc trudng hgp ddc biet mgt cdch hi thdng; (3) 77m kiem vd du dodn quy ludt; (4) Hinh thdnh gid thuyet; (5) Kiim chimg gid thuyit (vdi cac trudng hgp dac biet); (6) Tdng qudt hoa gid thuyet; (7) Xdc minh gid thuyit tdng qudt thi budc thii 4 {Hinh thdnh gid thuyet) la quan trpng va xuat hien thudng xuyen nhat trong bai lam cua HS. Day rd rang la nhiem vu ciia ngoai suy. Mpt sd cau hoi dugc chiing tdi dat ra: Lieu ngoai suy cd tham gia vao cac boat dgng kham pha quy luat day sd? Neu cd thi ngoai suy dugc the hien d dau trong qua trinh nay?
t^uay trd lai tim hieu cac nghien ciiu ve suy luan ngoai suy cda Peirce dac biet la d giai doan thii 2 (tir nam 1878 trd ve sau), chiing tdi tim thay mdt chi din cho cau tra ldi, do la den nam 1901, Peirce bat dau sii dung thuat ngii "ngoai suy" nham chi den
"sy khdi dpng dau tien nhat de dua ra mgt gia thuyet" (Peirce, [65, 6.525]). "Ngogi suy chi dan thudn Id bu&c khdi ddu. Nd la budc ddu tien ciia suy ludn trong khoa hgc, trong khi quy ngp la buac kit ludn sau ciing" (Peirce, [65, 7.218]). Chung tdi cung phat hien dugc mpt so diem khac biet sau day giiia ngoai suy va quy nap qua qua trinh khao cuu cac lai lieu lien quan:
Muc dich cua ngoai suy la dua ra mdt gid thuyet nham giai thich cho nhung gi dugc quan sat [7]. Muc dich cua quy nap nham tdng qudt hda mgt tinh chdt tir viec quan sat linh chat dd trong nhihig trudng hgp rieng.
Quy nap "cho thiy su tin tai ciia mdt hien tugng ma chung ta da quan sat trong nhung trudng hgp luong ty trudc dd", va "xu hudng nay khdng phai la cac su kien mdi" ([1], tr. 234), trong khi ngoai suy "de xuat mgt dieu gi do ma thudng la chung ta khdng thi quan sal mot each tryc tilp" ([8], 2.640). Ngoai suy la loai suy luan duy nhit lao ra cac tri thiic mdi cua ngudi hpc. Ket luan cua quy nap chac ehan hon ngoai suy, nhung it sang tao hon.
Quy nap chi ra sy phat trien ciia xu hudng dugc du doan cho nhiing quan sat xa hon, ngoai suy khong (true tiep) quan tam din nhung quan sat xa hon sau do ma chi hudng din muc dich li giai cho chinh trudng hgp dang xay ra. Noi each khac, ngoai suy bit diu khi cd mdt quan sat gay ngac nhien thuc diy viec tao ra mpt gia thuyet de giai thich d giai doan dau lien nhat, hoac lam hep bdt miln cac gia thuylt cd the xay ra.
Quy ngp chi bit dau van hanh khi da cd gia thuylt tir ngoai suy, bang each kiem tra gia thuylt thdng qua cac trudng hgp cu thi. Quy nap khdng hi tao ra bit ki cac y tudng co ban ban diu nao [8].
109
TAP CHI KHOA HOC DHSP TPHCM Sd 9(75) nam 2015
Nhiing phan tich tren cho tbay chuc nang va kit qua cua suy luan ngoai suy va quy nap la hoan toan khac nhau. Tu> nhieru viec phan biet hai loai suy luan nay irmig qua trinh kham pha cac quy luat day sd trd nen phiic tap hon tbeo chiing tdi bcri hai Ii do sau. Thir nhit, Deutscher [3] cho rang phep qin' nap gin liln vdi \ iec tdng qua! boa mgt thudc tinh hay mgt mdi quan he tir it nhit hai trudng hgp cu the cho mgt ldp toan bd cac ddi tugng, cdn phep ngoai su\ ddi hdi mdt bien ddi dot bien co tinh khai niem tu trudng hgp da cho den mdt gia thuyet co tinh giai thich. Noi each khac, ini thi ciia ngoai suy dugc tan dung khi dua ra gia thuyet chi dua tren mot quan sat don le (ho|k:
mot so quan sat cd lien quan den nhau nhimg khong nhdt thiet tirang tu nhau% trong khi qu\ nap can phai dya tren mdt sd lugng nao do cdc quan sdt tirang tu nhau. Thu hai. gia thuyet cua ngoai suy thudng ta phat bieu dya tren mdi quan he nguyen nhdn - he qud, va ket luan ciia quy nap la mpt phat bieu mang tinh tong qudt hoa. Tuy nhtea khi kham pha quy lual day sd, gia thuyet ban diu dugc de xuit phin Idn la dl li giai cho mgt vdi trudng hc^ da dirge cho san chir khdng chi mgt tnidng hgp, \a. gia thiwlt nay thudng bi nhim lan vdi ket luan ciia suy luan quy nap do nd co the duac tdng qudt hoa. Nhu \ a>. trong qua trinh kham pha quy luat day sd. \ iec de xuat gia thi^et ban diu nhat \ e qm' luat la cdng viec ciJa ngoai suy. nhung [diat bieu cudi ciing nhim tnig quat hda cua qi^- luat dugc khing dinh bdi quy nap. thdng qua kiem chung vdi cac trudng hgp thuc nghiem.
Chung tdi ciing tim tha> mgt quan diem tuong ty trong nghien cuu ciia Becker &
Rivera [3] khi cac tac gia quan sat va phdng van qua trinh suy luan cua 42 gjao vien (G\') toan trong luc giai qi^et cac nhiem va lien quan den tdng quat hda quy luat bac nhit Tren co sd qu> trinh kham pha cac quy luat ham so bac nhit bing ngoai suy-quj' nap dugc de xuat bdi Becker & Rivera [3] va md hinh suy luan quy nap gdm bay budc ciia ciia Canadas & Castro [4], chung tdi xay dung quy trinh li thiQ'et de kham pha quy luat day sd gdm 5 budc d hinh 2:
thap dii , beu VOl 1 haus S5
N G O A I SV\
<dexQal sia thayet ••; q i j - _^_ rac dira deii
' cac io liaii2 55 bicT •
^^
i QV\ X A P
1 m a roDg v a k i e m cfaDOg q n y t i c voi cac so bans c.i biet \3 mor vai>6 hang chua biet o vi tri gOD i
^
-j Bac bo
Q l ' ^ ' N . \ P i m o T o n g quy t i c cho c i c so . > bang choa b i l l d \"i tri xa va ^o bang rong quat-
V
Qny tac TQH chm^en ket qua cua z:n duiyet agoat
^u>"-qu>" tap thanh bieu
±irc dai so mo ra qiiy hiai day
Hmh 2. Quy trinh khdm phd quy ludt ddy sd bdng st/y ludn ngoai suy-quy nop
110
TAP CHl KHOA HOC DHSP TPHCM Truffng Thi Khanh Phuang
Minh hoa cho cac birac cua quy trinh se dugfc trinh bay cu the ngay sau day qua mot bai toan duac su dung trong thirc nghiem ciia nghien cuu nay.
2.4. Cdc phuong dn ngoai suy de khdm phd quy lugt day sd
Thirc nghiem dupc tidn hanh tren 81 HS thuoc hai lop 10 cua Truong THPT Le Loi, thanh pho Dong Ha, Quang Tri nhjm khao sat cac gia thuySt ngoai suy ma HS di xuit va quy trinh ma cac em thuc hien d6 Idiam pha ra quy luat day s5. HS dupc yeu ciu tra ldi hai bai toan: Hlnh chir Z va Hinh chO S. Hai bai toan dupc dua ra cho HS vpi boi canh hoan toan gifing nhau, BDTQ mo ta cho cac s6 hang cu (ai ciia mSi bai cung tuong tu nhau, nhung ham so mo ta quy luat bai Hlnh elm Z la mpt ham bac nhSt va bai Hlnh chits la ham bac hai.
CAU HOI 1. HINH CHtr Z
Nam su dung nhung tdm bia hinh vuong gi6ng het nhau dS thiSt k6 mau hinh chii Z trang tri cho bu6i tiec sinh nhat vai cac kich ca khac nhau. Dudi day la minh hoa mlu hinh chu Z ma Nam da thiSt ki vai ba kich ca tuong iing.
[
Co 1 Cd 2 Cd 3
Khi kich cd mau hinh chu: Z-tang len, se can chuan bi nhilu tam bia hinh vudng hon.
a) Em hay de xuat mdt quy tac giiip Nam tim sd tam bia hinh vudng can chuan bj cho mau hinh chii Z vdi cd bing gia trj n bat ki.
b) Md ta rd rang lam the nao em tim ra dugc quy lac do. Em cd the diing hinh ve, lap bang sd lieu hay dien dat bang ldi.
CAU HOI 2. HINH CHU'S
Binh sir dung nhiing tam bia hinh vudng gidng het nhau de thiet ke mau hinh chu S trang tri cho hdi trai vdi cac kich cd khac nhau. Dudi day la minh hoa mau hinh chii S ma Binh da thiet ke vdi ba kich co tuong img.
C d l Cd2 Co 3
TAP CHI KHOA HOC DHSP TPHCM Sd 9(75) nam 2015
Khi kich cd mau hinh chii S tang len, se cin chuin bi nhilu tim bia hinh vudng hon.
a) Em hay de xuat mot quy tac giiip Binh tim sd tim bia hinh vudng can chuan bi cho mau hinh chu S vdi cd bang gia tri n bat ki.
b) Md ta rd rang lam the nao em tim ra dugc quy tac dd. Em cd the dung hinh ve, lap bang sd lieu hay dien dat bang ldi.
Do sy tuong tu vl mat ban chat ciia hai bai loan tren, trong bai bao nay chiing toi chi trinh bay phan tich tien thyc nghiem dua tren quy trinh ma chiing toi d§ de xuat cho bai Hinh chic Z:
Budc 1. Quan sat, thu thap dir lieu cho cac trudng hgp cho san.
HS quan sat cac BDTQ md ta Hlnh chit Z cd 1, cd 2, cd 3, thu thap dii lieu bing each dem de co dugc sd lam bia ciia Hinh chir Z cd 1, cd 2, cd 3 la 5, 8, 11.
Bu&c 2. De xuat gia thuyet ngoai suy la mgt quy tac mang tinh tham do nham li giai cho sy xuat hien theo quy luat cua cac trudng hgp cd san. Quy tac n^y dugc phat hien dya tren viec to chirc, he thdng hda dir lieu sd (chiing tdi ggi la cac phuong an ngoai suy So hgc) hay khai thac cau tnic cua BDTQ md ta day Hinh chir Z theo cac each khac nhau nham lam xuat hien lap lai mpt sd dac trung nao do giQa cac trudng hgp cho san (chiing tdi ggi la cac phuong an ngoai suy Hinh hgc). Sau day la minh hoa mdt sd phuong an ngoai suy Sd hgc:
(1) Quy tdc de quy:
Bang 3.1. To chuc du lieu theo phuong an De quy
Kich Ctf (/i) 1 2 3
So tam bia (a„ ) 5 8 11
Phirtnig an ngo^i suy 5 8 = 5 + 3 11 = 8 + 3
Vdi each td chiic dii lieu nhu trong Bang 3.1, gia thuylt ngoai suy:
*^/,+i ='3'„ + 3,a, = 5 vdi n = l,2.
(2) Dodn vd Thic:
Bang 3.2. To chiic dii lieu theo phuong an Dodn vd Thir Kich ccf («)
1 2 3
So tam bia (a„) 5 8 11
Phirong an ngoai suy 5 = 3.1 + 2 8 = 3.2 + 2 11 = 3.3 + 2
112
TAP CHl KHOA HQC OHSP TPHCM Truffng Thi Khanh Phuffng
Vcri each t6 chiic dO lieu nhu trong Bang 3.2, gia thuyet ngoai suy: cac so hang ciia day Hlnh chir Z thoa a„ = 3n + 2 voi n = l,2,3.
(3) Cpng don:
Bdng 3.3. To chiic du lieu theo phuang an Cong don Kich CO" (n)
1 2 3
So tam bia (a„) 5 8 11
Phvong an ngoai suy 5 8 = 5 + 3 = 5 + 1.3 11 = 8 + 3 = 5 + 3 + 3 = 5 + 2.3 Vdi each td chiic du lieu nhu trong Bang 3.3, gia thuyet ngoai suy:
a^ = 5 + ( « - l ) 3 vdi n = l,2,3.
Tiep theo chung tdi minh hga mgt sd phuang an ngoai suy Hinh hgc:
(4) Ghep hinh rai: Chia Hinh chir Z thanh ba phan (mdi phan dugc danh dSu bing mpt mau rieng biet). Sd tam bia t^o thanh Hlnh chic Z dugc tinh bing each liy sd tim bia theo timg mau va cgng lai.
C&l
• i
^
2 + 1 + 2
C»2
H j ^
• , « - as
\
3 + 2 + 3
Co-3
• 1^
"^
H^\4 + 3 + 4
Gia thuyjt ngoai suy: S6 t4m bia cua Hlnh chir Z ca n (« = 1,2,3)la:
J „ = ( « + l) + n + (n + l ) .
(5) Lam Iron hinh: B6 sung vao m6i Hlnh chir Z 4 tjm bia (dupc to mau) de tao thanh cac hinh chO nhat. S6 tdm bia trong m6i Hlnh chir Z bang s6 tim bia cua hinh cha nh^t dupc tao thanh trii di 4 t4m bia vua dupc bo sung.
Co-1
3(l + 2 ) - 4
Co-2
3 ( 2 + 2 ) - 4
C&3
I
3(3 + 2 ) - 4
113
TAP CHI KHOA HOC OHSP TPHCM S6 9(75) ndm 2015
Gia thuyet ngoai suy: Sd lim bia cua Hinh chir Z cd n (« = 1,2,3)la:
a„=3(n + 2)~4.
(6) Ghep hinh chdng: Tudng tugng cac Hinh chic Z dugc tao thanh bang each ghep chdng len nhau cac Hinh chic Z cd 1, vdi sd tam bia ciia Hinh chic Z cd 1 la 5 tim.
Khi dd, can phai trir di cac lim bia bi tinh hai lan do chiing bi ghep chdng len nhau (la cac tam bia co danh dau X).
C f f l
r
1
l . ( 5 ) - 0
€9 2
r 1 L
~tat 1
2.(5)-1.2
Cff3
[^ i 1 m 1
+ rL+
fM 1 E3 1
3.(5)-2.2
Gia thuyet ngoai suy: So tam bia cua Hlnh chij- Z ca n (« = 1,2,3) la:
j ^ = „ ( 5 ) - ( « - l ) 2 .
Bu&c 3: M& rpng gid thuyet vira de xudt bdng siiy ludn quy nap vd time hien idem chung cho nhimg trudng hop chua biet a vi tri gdn nhdm khdng dinh hay bdc bo gid thuyet ndy.
Gia thuyet ngoai suy dupc dk xuit a Bupe 2 chi mai nhim giai thich cho cac truong hop cho san. De de xuat quy tac cho truemg hpp tong quat bang suy luan quy nap, cac quy tic o Buoc 2 cin dupc mo rpng cho cac truemg hop chua biSt (n = 4,5,6...)va kiem chiing. Gang nhieu truong hpp dupc kiem ehiing ia diing thi tinh CO II ciia gia thuyet ngoai suy eang dupc ciing co. Tuy nhien nlu co mpt truang hpp sai thi gia thuyet can dupc loai bo va HS quay tra lai Buac 2.
Buffc 4: Md rpng quy tdc cho tnrdng hop long qudt.
So vai gia thuyet ngoai suy dupc di xuit a Buoc 2, gia thuySt ngoai suy-quy nap a Buac 4 da dupc ma rpng va kiem chiing tinh diing din cho nhung truang hpp chua biet, sau dd tfing quat hoa len thanh mpt quy tic dl tinh sfi tim bia cho Hlnh chU Z voi CP n bat ki. Cu the la quy tic a„^, =a„+3,a, =5,n = 2,3,4... dupc ma rong va tfing quat hoa tir phuang an ngoai suy (1) hay quy tic a„ =3n + 2.V« = 1,2,3... dupc mo rfing va tfing quat hoa tir cac phuang an ngoai suy (2), (3), (4), (5), (6). 6 buac nay, HS CO the nhan ra viec thuc hanh quy tic a^, =a„+3,a, =5,n = 2,3,4... cho d5y Hlnh cha Z se gap kho khan khi gia tri n dupc yeu ciu la mpt sfi kha Ion, ching han tim so tim bia can sii dung cho Hinh chir Z co 500. Do do, HS co thi quay tro Iji Bufic 2 de tim kiem mpt quy tic khac giiip viec giai quyet van de hieu qua ban.
114
TAP CHl KHOA HOC DHSP TPHCM Truffng Thi Khdnh Phuong
Bu&c 5: Kit ludn vi quy ludt cua ddy sd
HS dua ra kit luan cufii cimg vl quy luat ciia day sfi sau khi da kilm chung tinh diing din va danh gia hieu qua ciia gia thuylt ngoai suy-quy nap trong viec tim kilm mpt sfi hang ciia d§y sfi a vi tri bat ki.
Voi cac kit qua thu thap dupc tir bai lam cua HS, chung tfii tiln hanh phan loai cac phuong an ngoai suy theo timg pham trii va ti ti le cau tra loi dung trong mfii pham tril vai hai nhiem vu Hlnh chir Z va Hlnh chir S. Rieng nhQng HS khong dua ra gia thuylt hoac dua ra gia thuylt vl quy tic tfing quat nhung khfing li giai, hoac li giai hoan toan khong hpp li se dupc xIp vao pham trii "Khfing xac dinh dupc".
Bang 3.4 va 3.5 cho thiy su phan bfi cac phuong an ngoai suy dupc HS thi hien theo timg pham trii va ti le cau tra loi dung trong mfii pham trii vai hai nhiem vu Hlnh chir Z va Hlnh chic S.
Bdng 3.4. So luong cdc phuong dn ngogi suy theo timg pham trit
Nhifm vu So hoc Htnh cha z\ 39 Hlnh chu S\ 9
Hinh hoc 15 16
Boan va thir 3 4
Khong xac dinh dirorc 24 52 Bang 3.5. Sd luang vd ti le cdu trd ldi dung trong mdi phgm trii
Nhiem vu Hlnh cha Z
Hlnh cha S
Sah9C 19 (48,7%)
0 (0%)
Hinh hoc 12 (80%)
11 (68.8%)
Doan va thir 1 (33,3%)
0 (0%)
Khong xac dinh diro-c 1 (0,04%)
0 (0%) Mgt sd ket luan dugc nit ra tir thuc nghiem qua quan sat bai lam cua HS va cac sd lieu dugc thdng ke trong Bang 3.4. va Bang 3.5:
Thii nhat, ti le HS dua ra cac phuong an ngoai suy thudc pham trii "Khdng xac dinh dugc" d bai Hinh chir Z la 25/81 va bai Hinh chir S la 52/81. Ti le nay cho thiy viec dl xuit mgt gia thuylt ngoai suy cd li cho bai loan quy luat ham bac hai gap trd ngai nhilu hon so vdi quy luat ham sd bac nhit. Vdi bai Hinh chir Z, trong sd 39 phuong an ngoai suy So hgc, co 19 HS (gin 50%) cd th6 dua ra dugc quy tac tdng quat diing trong khi con sd nay d bai Hinh chir S la bing 0 mac dii hau het HS da nhan thay gia tri sai khac giQa hai sd hang lien tiep trong bai Hinh chit S la mgt day theo quy luat cip sd cgng. Dieu nay cho thay cac phuong an ngoai suy Sd hpc (chang ban viec su dung quy tic d? quy) khong con phat huy hieu qua khi kham pha cac day sd theo quy luat ham so bac hai. Hon nua, ti le HS dua ra quy tac diing tir cac phuong an ngoai suy
TAP CHi KHOA HOC OHSP TPHCM s6 9(75) ndm 2015
Hinh hgc ludn cao hon trong ca hai bai loan cho thiy hieu qua cua nhiing phuong an ngoai suy nay mang lai.
Thii hai, vdi ngoai suy Sd hgc, chi cd mdt quy tic diing dugc HS dira ra cho bai Hinh chit Z la 5 + 3 ( n - l ) va khdng cd kit qua nao cho bai Hinh chir 5 thi vdi ngoai suy Hinh hgc, cd ba quy t ^ tuong duong la 3/7 + 2;2(/7 + l) + n;3(n + 2) —2^ cho bai //i/iA cAi? Z va ba quy tic (l + n ) + ( l + n) + «";2 + (n + 2)rt;(n + 2)"-2(n+l)chobai Hinh chu S. Ket hgp vdi quan sat bai lam cua HS, chiing tdi cho ring ket qua tren co dugc la do cac BDTQ cung cap cho HS nhieu each nhin khac nhau ve sy phat trien ciia qu> luat hon so vdi viec sOr dung don thuan cac dir lieu so.
Thii 3. nhung HS stir dung ngoai suy Sd hgc thudng trinh bay giai thich cho quy tac tdng quat dugc de xuat bang each kiem chiing rang quy tac nay diing vdi cac trudng hgp da biet ( H = 1,2,3) trong khi nhimg HS tien hanh ngoai su\- Hinh hgc cd the md la mgt sd hang tdng quat thdng qua BDTQ, nghia la cac em cd the nhan ra dugc mdi lien ket mang tinh quy luat giira sd hang \a vi tri ciia nd trong day sd mdt each ddc l ^ vdi cac sd hang khac (Hinh 3).
dat bdng ldi. a v6l
•:::m::::::::::mi:::z:::::::^::.m::^ii^. ... T:;::::::::;::
::i::f::::::t;kz:z^ -.rir-m ^ ""':"="::::
* . - , .if..-.^ , - . . .
jifi3oilnJfC<L .4?.taniliuaj«ti)MJL _. -- -
.«Sxoi.Ail3. -..- - Hinh 3. Ngogi suy Hinh hgc cho bdi Hinh chii Z
Thii 4, mgt sai lam kha phd bien xuit hien trong cac phuong an ngoai su} Sd hgc la HS chua that sy hieu y nghia cda bien sd n dan den cac quy tic sai mac dii chiing co the trich xuat ra diing day sd ciia bai loan. Chang ban, trong quy tic n" +1, Vn > 2 ma HS dua ra cho bai Hinh cfnr S (Hinh 4) thi bien n khdng mang y nghia dai dien cho kich cd ciia Hinh chir S, hay cac quy tac de qu\ dugc dien dat dudi dang n+3 trong bai Hlnh chir Z (bien n la kich cd cua Hinh chir Z nhimg trong cdng thiic nay nd dai dien cho sd hang diing d trudc do).
::cJ^:i:s:::?::::::::;:
.iQ..A.-:\..?T.S..-:i.n.^.2.
a.2..: ~ 1—<Q.o. >.-.—ii— >-- v^: f
..yj.'-r.l .-.A.O..~?.yxj=L.$ ;jCs....c«ta-..tU--c>u^..-L>.-...n^.H-
Lfl' 4 ^ o . . . V"ia.>.Z i*. £ M .xC-:--ir^.-^ < 3 • . H & n - 4 ^
Hinh 4. Ngogi suy Sd hgc cho bdi Hinh chu S
116
TAP CHi KHOA HOC OHSP TPHCM Truomg Thi Khdnh Phuang
Cudi cimg, qua quan sat bai lam cda HS, chiing tdi hau het cac em deu chi dua ra gia thuylt ngoai suy dl li giai cho cac trudng hgp dugc cho sin va sau dd tdng quat hda len ma khdng kilm chung quy tac cho cac trudng hgp chua bi6t.
3. Ket luan
Trong bai bao nay, chiing toi da phan tich co sd li thuyet de lam ro sy khac nhau giiia hai loai suy luan quy nap va ngoai suy, d^c biet la trong ho^t dpng tong quat hda quy luat day sd. Chiing toi cijng de xuat quy trinh li thuyet the hien sy kSt hgp chat che va hd trg lan nhau ciia suy luan ngo^i suy va quy nap trong viec kham pha va kiem chung mdt gia thuyet de cudi cung di den quy tac tdng quat. Nhihig ket qua thyc nghifm cho thiy: (1) HS thudng bo qua giai doan kiem chiing gia thuylt ngoai suy cho cac trudng hgp TQH gan va TQH xa ma de xuat ngay quy lac tdng quat khi thay quy tic diing cho cac trudng hgp da biet; (2) BDTQ md ta quy luat day sd cung cap mpt co sd tit vl tinh CO li cho gia thuyet ngoai suy qua do ban chS dugc cac sai lam (dac biet la cac sai lim vl y nghTa ciia bien sfi) va giup HS cd nhieu hudng tiep can khac nhau ddi vdi quy luat day sd (dac biet la cac day sd theo quy luat ham sd bac hai, dong thdi hi trg quy nap trong viec tdng quat boa cac gia thuyet nay. Mdt bang phan loai cac mire do ciia suy l u ^ ngoai suy dugc HS the hien khi giai quylt cac nhiem vu tdng quat hoa quy luat day sd dugc md ta bing bieu dien true quan da dugc chiing tdi thiet ke de cung cip nhiing phan tich sau sic hon ve mat thyc nghiem, ndi dung nay d nghien ciiu chiing tdi se trinh bay tilp theo.
'National Council of Teachers of Mathematics (2000), Principles and Standards in Mathematics, NCTM, USA
- Polya (1954), Pattems of Plausible Inference, pp. 158-160,
TAI LIEU THAM KHAO
1. Abe, A. (2003), "Abduction and analogy in chance discovery". In Y. Ohsawa & P.
McBurney (Eds), Chance Discovery, pp. 231-248, New York: Springer.
2. Billings, E. M. H. (2008), "Exploring generalisation through pictorial growth pattems", In C. E. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics (Seventieth Yearbook), pp. 279 - 293, Reston,VA: NCTM.
3. Becker, J., & Rivera, F. (2007), Abduction in pattern generalization. Proceedings of the 31 St conference of the International Group for the Psychology of Mathematics Education, Vol (4), pp. 97-104, Seoul, Korea: Korea Society of Educational Studies in Mathematics.
117
TAP CHi KHOA HOC OHSP TPHCM S6 9(75) nam 2015
4. Canadas, M.C. & Castro, E. (2009), "Using a model to describe students' inductive reasoning in problem solving", Electronic Journal of Research in Educational Psychology, 1 (17), pp. 261 - 278.
5. Eco, U. (1983), "Homs, hooves, insteps: Some hypotheses on three types of abduction". In U. Eco & T. Sebeok (Eds.), The sign of three: Dupin, Holmes, Peirce, pp. 198-220, Bloomington, IN: Indiana University Press.
6. Josephson, J. & Josephson, S. (1994), Abductive Inference: Computation, Philosophy, Technology, New York: Cambridge University Press.
7. Magnani, L. (2005), "An abductive theory of scientific reasoning". Semiotica, 153(1- 4), pp. 261-286.
8. Peirce, C. S. (1960), Collected Papers, Cambridge, MA: Harvard University Press.
9. Reid, D. (2002), "Conjectures and refutations in grade 5 mathematics". Journal for Research in Mathematics Education, 33(1), pp. 5-29.
(Ngdy Ida soan nh$n dugc bai: 07-02-2015, ngay phan bien d^nh gid' 10-9-2015;
ngdy chSp nhdn dSng: 24-9-2015)
